DIRECTIONAL DO-NOTHING CONDITION FOR THE NAVIER-STOKES EQUATIONS

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations discretized by finite elements is the "do-nothing" condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a "directional do-nothing" condition. In contrast to the usual "do-nothing" condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.

[1]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[2]  Giovanni Paolo Galdi,et al.  Linearized steady problems , 1994 .

[3]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[4]  Roland Becker,et al.  A finite element pressure gradient stabilization¶for the Stokes equations based on local projections , 2001 .

[5]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[6]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[7]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[8]  R. Rannacher,et al.  Benchmark Computations of Laminar Flow Around a Cylinder , 1996 .

[9]  M. Feistauer,et al.  On non‐stationary viscous incompressible flow through a cascade of profiles , 2006 .

[10]  L. Stupelis.  Some problems of vector analysis , 1998 .

[11]  Eduard Feireisl,et al.  Dynamics of Viscous Compressible Fluids , 2004 .

[12]  Malte Braack,et al.  Finite elements with local projection stabilization for incompressible flow problems , 2009 .

[13]  P. Gresho.  Some current CFD issues relevant to the incompressible Navier-Stokes equations , 1991 .

[14]  Piotr Bogusław Mucha,et al.  On Navier–Stokes Equations with Slip Boundary Conditions in an Infinite Pipe , 2003 .

[15]  P. Mucha,et al.  On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion , 2000 .

[16]  P. Fabrie,et al.  New efficient boundary conditions for incompressible Navier-Stokes equations : a well-posedness result , 1996 .

[17]  Giovanni Paolo Galdi,et al.  Nonlinear steady problems , 1994 .

[18]  G. Burton.  Sobolev Spaces , 2013 .